Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{9k^3 - 324k}{-k^2 - 3k + 18}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {9k(k^2 - 36)} {-1(k^2 + 3k - 18)} $ $ y = -\dfrac{9k}{1} \cdot \dfrac{k^2 - 36}{k^2 + 3k - 18} $ Next factor the numerator and denominator. $ y = - 9k \cdot \dfrac{(k + 6)(k - 6)}{(k + 6)(k - 3)}$ Assuming $k \neq -6$ , we can cancel the $k + 6$ $ y = - 9k \cdot \dfrac{k - 6}{k - 3}$ Therefore: $ y = \dfrac{ -9k(k - 6)}{ k - 3 }$, $k \neq -6$